Polynomial stability of exact solution and a numerical method for stochastic differential equations with time-dependent delay

Polynomial stability of exact solution and modified truncated Euler-Maruyama method for stochastic differential equations with time-dependent delay are investigated in this paper. By using the well known discrete semimartingale convergence theorem, sufficient conditions are obtained for both bounded and unbounded delay $\delta$ to ensure the polynomial stability of the corresponding numerical approximation. Examples are presented to illustrate the conclusion.